Question: Determine how many solutions exist for the system of equations. ${12x-2y = -20}$ ${-6x+y = 10}$
Solution: Convert both equations to slope-intercept form: ${12x-2y = -20}$ $12x{-12x} - 2y = -20{-12x}$ $-2y = -20-12x$ $y = 10+6x$ ${y = 6x+10}$ ${-6x+y = 10}$ $-6x{+6x} + y = 10{+6x}$ $y = 10+6x$ ${y = 6x+10}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 6x+10}$ ${y = 6x+10}$ Both equations have the same slope and the same y-intercept, which means the lines would completely overlap. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Since any solution of ${12x-2y = -20}$ is also a solution of ${-6x+y = 10}$, there are infinitely many solutions.